Finding the extremum of function z = x square + XY + y square - 3x-6y

Finding the extremum of function z = x square + XY + y square - 3x-6y

It's just finding partial derivatives
Z’｜x=2x+y-3
Z’｜y=x+2y-6
Let Z '︱ x = 0, Z' ︱ y = 0,
The combination equation is x = 0, y = 3
That is, (0,3) is the stationary point of Z,
So the extremum is f (x, y) = - 9

Let f (x) = the square of ax3-3x (a ∈ R), and x = 2 be the extremum of y = f (x). (1) find the value of real number a, and find the monotone interval of the function. (2) find the monotone interval of G (x) = e square, x = ex times f (x)

（1）
f'(x)=3ax^2-6x
Because x = 2 is the extremum of y = f (x)
So f '(2) = 12a-12 = 0
So a = 1
Now we know that f (x) = x ^ 3-3x ^ 2
There are two extreme points: x = 0 and x = 2
x

Find the extremum of function f (x, y) = XY (x square + y square - 1)

Unconditional extremum problem
The partial derivatives of X and y are obtained respectively, and the point (x0, Y0) with partial derivative = 0 is obtained
Then, calculate the second-order partial derivatives FXX (x0, Y0) = a, fxy (x0, Y0) = B, FYy (x0, Y0) = C respectively
B ^ 2-ac > 0 (x0, Y0) is not an extreme value
B ^ 2-ac0 minimum; a

Finding the extremum of function f (x, y) = XY + 1 / x + 1 / Y

Finding partial derivative of x = Y-1 / x ^ 2 = 0
Finding partial derivative of y = X-1 / y ^ 2 = 0
Get x = 1, y = 1
So the extremum is 3
Find a second-order partial derivative, you can see is a minimum

X & # 178; - 4x + 1 = 0, X & # 178; + 3x + 1 = 0, use formula to solve equation
emergency

The first X & # 178; - 4x + 4-3 = 0
（x-2）²-3=0
（x-2）²=3
X-2 = positive and negative root sign 2
X = 2 plus minus radical 2
The second X & # 178; + 3x + 9 / 4-5 / 4 = 0
（x+2/3）²-5/4=0
The steps to solve the equation are the same as above

It is known that the positive scale function y = KX passes through the point P (1,2), as shown in the figure. (1) find the analytic expression of the positive scale function; (2) translate the image of the positive scale function to the right four units, write out the coordinates of the image p 'and o' of the point P and the origin o under this translation, and find the analytic expression of the straight line after translation

(1) Since the point P (1,2) is on the straight line y = KX, K · (1 = 2) leads to k = 2. The analytic formula of this positive proportional function is y = 2x. (2 points) (2) P ′ (5,2), O ′ (4,0) (3 points). Let the analytic formula be y = KX + B (K ≠ 0), and substitute P ′ (5,2), O ′ (4,0) to get 5K + B = 24K + B = 0. The solution is k = 2B = - 8. (5 points). Therefore, the analytic formula is y = 2x-8. (6 points)

Li Ming and Wang Yun are facing each other from a and B. if they start at the same time, they will meet in 80 minutes. If Li Ming starts 60 minutes later and Wang Yun starts again, they will meet in 40 minutes. How many hours does it take Li Ming and Wang Yun to complete the AB journey alone?

Suppose that it takes Li Ming x hours to complete the whole journey, and the distance is 1. The equation can be listed: 60 + 4060 × 1x + 4060 × (1 △ 8060-1x) = 1. The solution is: x = 2. After the test, x = 2 meets the meaning of the problem. 1 △ (1 △ 8060-1x) = 4. Answer: it takes Li Ming and Wang Yun 2 hours and 4 hours to complete the journey alone

In the plane, the distance from the moving point P to the y-axis is less than 1 to the fixed point F (0,1). The trajectory equation of the moving point P is obtained
RT

In the plane, the distance from the moving point P to the y-axis is less than 1 to the fixed point F (0,1), that is, the distance from P to y = - 1 is equal to the distance to the fixed point F (0,1). The trajectory equation of the moving point P defined by the parabola is x ^ 2 = 4Y

Factorization 1-A ^ 4

1-a^4 =(1+a²)(1-a²)
=(1+a²)(1+a)(1-a)

There are five Saturdays and four Sundays in October of a certain year. October 1 of this year is ()
A. Monday B. Tuesday C. Wednesday D. Thursday

October has 31 days, and 31 = 4 × 7 + 3, so this month has 4 weeks and 3 days. We can use the hypothetical method to calculate the first Saturday of this month: (1) if October 1 is a Saturday, then October 2, 9, 16, 23 and 30 are all Sundays, and there are 5 Sundays, which is not consistent with the meaning of the title; using the same method, we can calculate that October 2 is not a Saturday (2) If October 3 is a Saturday, then October 4, 11, 18 and 25 are Sundays, which are exactly four Sundays. If you push back, you can know that October 1 is a Thursday